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Out of 200 students who are trying to improve their vocabulary, 120 students read newspaper H, 50 read newspaper T and 30 read both newspaper H and T. find the number of students

i) who read H but not T _____

ii) who read T but not H _____

iii) who don't read any newspaper _____

If $co{t}^{2}x=cot(x-y).cot(x-z)$ where $x\ne \pm \frac{\pi }{4}$, then $cot2x=$

If $x$ satisfies the inequality ${\log }_{x+3}(x^2-x)<1,$ then

Which among the following relations on $Z$ is an equivalence relation

Solve the integral I = ${\int }_0^{\pi }si{n}^{2}xdx$

If A, B, C are actual positive angles such that A + B + C = $\pi$ and cot A cot B cot C = k then

If $G$ is the geometric mean of $x$ and $y,$ then $\frac{1}{G^2-x^2}+\frac{1}{G^2-y^2}=$

The solution set of the inequality ${\log }_3\left(\right(x+2\left)\right(x+4\left)\right)+{\log }_{1/3}(x+2)<\frac{1}{2}{\log }_{\sqrt{3}}7$ is

Let $S_1,S_2,S_3$ and $S_4$ be four sets defined as
$S_1=\{x:x\in Z\text{and}{\log }_2|4-3x|\le 2\}$
$S_2=\{x:x\in Z\text{and}|1-\frac{|x|}{1+|x|}|\ge \frac{1}{3}\}$
$S_3=\left\{x:x^2-3x+2\text{sgn}\left(x\right)=0\right\},$ where $\text{sgn}\left(x\right)$ represents the signum function.
$S_4=\left\{\right(x,y):x,y\in Z,x^2+y^2\le 4\}$.

$\text{List I}$ has four entries and $\text{List II}$ has five entries. Each entry of $\text{List I}$ is to be correctly matched with a unique entry of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& n\left(S_1\Delta S_2\right)& \text{(P)}& 9\\ \text{(B)}& n((S_1\times S_2)\cap (S_2\times S_1))& \text{(Q)}& 12\\ \text{(C)}& n(S_1\cap S_2\cap S_3^{\prime })& \text{(R)}& 36\\ \text{(D)}& n(S_4\times S_3)& \text{(S)}& 2\\ & & \text{(T)}& 0\end{array}$

Which of the following is the only CORRECT combination?

The number of terms in the sequence $3,7,11,\dots ,407$ is

A circle with centre at the origin and radius equal to a meets the axis of x and A and B. $P\left(\alpha \right)$ and $Q\left(\beta \right)$ are two points on this circle so that $\alpha -\beta =2\gamma$, where $\gamma$ is a constant. The locus of the point of intersection of AP and BQ is

$\underset{x\to 3}{lim}\left(\right[x-3]+[3-x]-x),\text{where[.]denote the greatest integer function, is equal to:}$

The polar form of $\frac{-\sqrt{3}}{2}-\frac{i}{2}$ (where i = $\sqrt{-1}$) is

If $p^{th}$,$q^{th}$,$r^{th}$ and $s^{th}$ terms of an A.P. be in G.P., then (p - q),(q - r),(r - s) will be in

If $n(\cup$)= 700, n(A) = 200, n(B) = 300 , $n(A\cap B$) = 100, then $n(A^{{}^{\prime }}\cap B^{{}^{\prime }}$)

If $\underset{x\to a}{lim}$[f(x) + g(x)] = 10 and $\underset{x\to a}{lim}$ f(x)=2, then find the value of $\underset{x\to a}{lim}$ g(x), provided the limit

exist

How many 5 digit even numbers can be made from the digits 0,1,2,3,4,5 if repetition is not allowed?

Find the coefficient of $x^{8}in(1+x{)}^8(1-x{)}^2$.

If G(x) = - $\sqrt{25-x^2}$ then $\underset{x\to 1}{lim}\frac{G\left(x\right)-G\left(1\right)}{x-1}$ has the value

If a,b,y are the altitudes of a $△ABC$ and 2s denotes its perimeter,
then $a^{-1}+B^{-1}+y^{-1}$ is equal to

The maximum value if 2 (a-x)(x + $\sqrt{x^2+b^2})$ is ( x , a, b $\in$ R )

If the coefficients of $r^{th}$ term and $(r+4{)}^{th}$ term are equal in the expansion of $(1+x{)}^{20}$, then the value of r will be

If $(p+q{)}^{th}$ term of a G.P. be m and $(p-q{)}^{th}$ term be n, then the \(p^{th}) term will be

If A (2, 1), B(-2, 3) and C(4, 5) are the vertices of a $△$ ABC, then find the equation of

(i) the altitude through B.

(ii) the right bisector of side BC.

$sin\left(2si{n}^{-1}\sqrt{\frac{63}{65}}\right)=$

Find the value of expression
$\sin (-\theta )+\cos (-\theta )+\sec (-\theta )$
$+\sin (\pi -\theta )+\cos (\pi -\theta )+\sec (\pi -\theta )$

Q(h, k) is the foot of the perpendicular of P(3, 6) on the line x - 2y + 4 =0.

If the slope of PQ is m, find $m^2$.

Let U be the universal set and $A\cup B\cup C=U$. Then {$(A-B)\cup (B-C)\cup (C-A)$}. Then U is equal to

In an equlilateral triangle, (circumradius) : (inradius) : (exradius) is equal to

If $tanx+tan(x+\frac{\pi }{3})+tan(x+\frac{2\pi }{3})=3,$ prove that $\frac{3tanx-ta{n}^{3}x}{1-3ta{n}^{2}x=1.}$

Or

If $sin\theta =nsin(\theta +2\alpha ),$ prove that $tan(\theta +\alpha )=\frac{1+n}{1-n}tan\alpha .$

The number of terms in the expansion of $(a+b+c{)}^n$

will be

Points (-2, -1), (4, 0), (3, 3) and (-3, 2) are the vertices of __

Evaluate $\underset{x\to \sqrt{10}}{lim}\frac{\sqrt{7-2x}-(\sqrt{5}-\sqrt{2})}{x^2-10}$

Or

Differentiate $x^{2}cosx$ by first principle.

The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

Find the number of positive real roots for equation 3 $x^5$ - 4 $x^4$ - 42 $x^3$ + 56 $x^2$ + 27x - 36 = 0. If all the roots of above given equation are real.

The coefficient of $x^5$ in the expansion of $(1+x{)}^{21}+(1+x{)}^{22}+........(1+x{)}^{30}$ is:

In a $△$ ABC, the tangent of half the difference of two angles is one - third the tangent of half the sum of the two angles. The ratio of the sides opposite the angles is

$1+2+3+...............+n$

If ${}^{56}{P}_{r+6}:{}^{54}{P}_{r+3}$ = 30800:1, then r =

If $S=4x^2-\sin x$, find $\frac{d^{2}S}{d{x}^2}$

The values of $x$ for which ${\log }_3(x+2)(x+4)+{\log }_{1/3}(x+2)<\frac{1}{2}{\log }_{\sqrt{3}}7$ holds, is

If $||x-3|+4|\ge 2$, then $x$ belongs to

The value of $x$, which satisfy the equation $3^{{\log }_{a}x}+2\cdot x^{{\log }_{a}3}=9$ is

Find the integral of the given function w.r.t - x

$y=\frac{si{n}^2\sqrt{x}}{2\sqrt{x}}$

If the line (x-y+1) + k (y-2x+4) = 0 makes equal intercept on the axes then the value of k is

The domain of the function $f\left(x\right)=2^{{\log }_{2}x}+\frac{2x+3}{x^2-4x+3}$ is

If S and T are two sets such that S has 21 elements T has 32 elements and $S\cap T$ has 11 elements, how many elements does $S\cup T$ have?

If $x+y+z=1$, $x,y,z>0$. Then greatest value of $x^2{y}^3{z}^4$ is